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# Hot air balloons

## by johnwest on Apr 21, 2010

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the science and maths behind balloon flight.. I took an interest in this after a balloon flight accross west auckland

the science and maths behind balloon flight.. I took an interest in this after a balloon flight accross west auckland

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## Hot air balloonsPresentation Transcript

• Hot Air Balloons The science and maths http:// johnwest.edublogs.org / Science Infoblog: A school science blog
• First a few useful definitions (fluids liquids and gasses)
• Fluids are substances that flow.
• By definition all liquids and gasses are fluids.
• Liquids are fluids that take the shape of the container they are stored in.
• Gasses are fluids that expand to fill up the volume of their container.
• Flotation and Archimedes
• Archimedes of Syracuse was a physicist, engineer inventor and astronomer in his spare time.
• …… and also probably the greatest mathematician ever.
• Archimedes was killed by a Roman soldier during the siege of Syracuse c. 212 BC
• Any object wholly or partially immersed in a fluid is buoyed up by a force equal to the weight of fluid displaced by the object (Archimedes Principle)
Archimedes on wikipedia
• Generating Lift With Hot Air
• Raising the air temperature inside the envelope reduces its density.
• The buoyant force according to Archimedes Principle is the weight of cold air displaced by the balloon.
• This is greater than the force of gravity on the heated air
• Hot air rises, its lighter than cold air
Gravity Buoyant force
• Balloons have to be big to lift. How hot is the air inside?
• It is the combination of size and temperature difference between the hot air in the envelope and the ambient air temperature that determines the amount of lift.
• If you want to design a balloon that will lift a basket with eight people in it…… how do you do the maths?
• The ideal gas equation is a good start
• What do we know about gasses
• Gasses expand in a predictable way when they are heated under a constant pressure ( Charles’s law )
• The volume of a gas decreases when it is put under pressure ( Boyle’s law ). If you double the pressure on a gas its volume halves. ( as long as the temperature doesn’t change)
• … .. And finally the volume of a gas only depends on the number of particles it contains (not so obvious)….see the next slide
• Counting atoms and molecules
• Air consists mostly of oxygen and nitrogen molecules
• They are too small to count individually
• Banks count out large numbers of coins by weighing them. They are bundled up in bags
• The bundle of molecules that chemists work with is called a mole .
• A mole of molecules contains 6x10 23 particles
• The number of moles is represented in equations with the letter n
• The Ideal gas equation
• Combing what we know about gasses (boyle’s Law, Charles’s Law and the number of particles present) leads us to the Ideal Gas equation
Increase p, V must decrease What happens to the gas when T increases? The gas constant,
• The Gas Constant
• If we measure pressure (p), volume (V), temperature (T in degrees absolute) and carefully weigh out an exact number of particles ( remember the bundle called a mole) then we can experimentally determine the value of R
or
• Using the ideal gas equation to calculate air densities
• Hot air rises because it is less dense than surrounding air
• We need to calculate air densities at different temperatures. We can rearrange the ideal gas equation to help us
• It is easier to do this if we fiddle with the gas constant but it does mean introducing another term. The Molar Mass.
• Molar Mass
• Chemists need to count accurately the numbers of particles in a chemical reaction otherwise there is no point to their chemical equations
• As said previously the bundle of particles they count with is called a mole (6.023X10 23 )
• A mole of hydrogen molecules weighs 2g
• A mole of oxygen molecules weighs 32 g
• A mole of water molecules weighs 18g
• If you look at a periodic table it is easy to see where those numbers come from
• The gas equation and air density (1)
• The density of air over a range of temperatures can be calculated using the gas equation.
• The number of moles of any gas present will be the mass of the gas divided by the mass of 1 mole. Now some maths.
• The gas equation and air density (2)
• The density of a gas is calculated by dividing its mass by the volume it occupies
• Air density is represented by the Greek letter rho.
• The gas equation and air density (3)
• We need to substitute and rearrange the equation a bit
Get rid of n Shift V to the other side
• The gas equation and air density (4) We can replace m/V with the air density and rearrange There is one last convenient trick
• The gas equation and air density (5). Using the specific gas constant
• The gas constant applies to any gas. We can tidy up the expression by using a new constant that is specific to a particular gas
Finally…
• The final equation for calculating air density
• R (specific) is a constant
• Pressure and temperature can be measured
• Balloon lift calculation
• P is the atmosperic pressure in Pascalls
• R is the specific gas constant for air
• T is the air temperature O K
• The lift experienced by cubic metre of air at 95 o C is 0.245Kg
• (ambient temp 15 o C)
For typical atmospheric conditions 1/0.25839 or 4.073 m 3 of balloon needed to lift 1 Kg.